Fluid Flow and Piping

Transcription

1 SECTION 17 Fluid Flow and Piping Few flow problems can be solved with an acceptable degree of accuracy when using equations designed to fit idealized applications. Flow regimes and associated pressure drops are complex phenomena and require complex equations to predict their relationships. For engineering design purposes, several empirical formulas have been developed to fit particular circumstances in predicting flow capacity and pressure drop. FIG Nomenclature A = pipe cross sectional area, ft 2 (A = πd 2 /4) c = sum of allowances for corrosion, erosion, etc., in., Fig C = design parameter used in Hazen and Williams formula, Eq C 1 = discharge factor from Fig C 2 = size factor from Fig d = internal diameter of pipe, in. d o = outside pipe diameter, in. D = internal diameter of pipe, feet E = pipeline efficiency factor (fraction) E = longitudinal weld joint factor from ANSI B31.3, Fig E = longitudinal joint factor from ANSI B31.8, Fig f f = Fanning friction factor f m = Moody friction factor (f m = 4.0 f f ) f n = single phase friction factor for Dukler calculation, from Eq f tpr = friction factor ratio for Dukler calculation, Fig F = construction type design factor used in ANSI B31.8, Fig F pv = volume correction for a non-ideal fluid due to compressibility from Eq /f f = transmission factor g = acceleration due to gravity, 32.2 ft/sec 2 g c = gravitational constant, 32.2 (ft lbm)/(lbf sec 2 ) h L = loss of static pressure head due to fluid flow, feet of fluid H = total energy of a fluid at a point above a datum, from Eq 17-1 H Ld = liquid holdup fraction (Dukler), Fig H Le = liquid holdup fraction (Eaton), Fig H Lf = liquid holdup fraction (Flanigan), Fig I L = liquid inventory in pipe, ft 3, from Eq L = length of line, feet L m = length of line, miles MW = molecular weight N x = Fig horizontal coordinate, ft/sec = Fig vertical coordinate, ft/sec N y N E = abscissa of Eaton correlation, Fig N Lv = liquid velocity number, from Eq N gv = gas velocity number, from Eq N d = pipe diameter number, from Eq N L = liquid viscosity number, from Eq P = pressure, psia P 1 = inlet pressure, psia P 2 = outlet pressure, psia P avg = average pressure, psia, from Eq P b = base absolute pressure, psia (ANSI 2530 specification: P b = psia) P i = internal design pressure, psig = pressure drop, psi/100 ft equivalent pipe length P e = elevation component of pressure drop, psi P f = frictional component of pressure drop, psi P t = total pressure drop, psi q = flow rate, gal./min Q = flow rate of gas, cubic feet per day at base conditions Q L = liquid volumetric flow rate at flowing conditions, ft 3 /sec Q g = gas volumetric flow rate at flowing conditions, ft 3 /sec Re = Reynolds number Re y = mixture Reynolds number for Dukler calculation, from Eq S = specific gravity of flowing gas (air = 1.0) S = allowable stress, psi, Fig S = specified minimum yield strength, psi, Fig t = thickness, in., Figs , t m = minimum required wall thickness, in., Fig T = absolute temperature of flowing gas, R T = temperature derating factor used in ANSI B31.8, Fig T avg = average temperature, R, [T avg = 1/2 (T in + T out )] T b = base absolute temperature, R (ANSI 2530 specification: T b = 520 R) V = single phase fluid velocity, ft/sec V sg = superficial gas velocity, ft/sec, from Eq V sl = superficial liquid velocity, ft/sec, from Eq V m = mixture velocity, ft/sec, from Eq P

2 FIG (Cont d) Nomenclature W = mass flow, lb/hr X A = Aziz fluid property correction factor (horizontal axis, Fig ) Y A = Aziz fluid property correction factor (vertical axis, Fig ) Y = coefficient found in Table , ANSI B31.3, Fig = average compressibility factor Z e = pipeline vertical elevation rise, ft ε = absolute roughness, ft λ = flowing liquid volume fraction µ e = single phase fluid viscosity, lb m /(ft sec) µ = single phase fluid viscosity, cp µ g = gas viscosity, cp Z avg Bernoulli s Theorem The Bernoulli Theorem 1 is a mathematical derivation based on the law of conservation of energy. This theorem states that the total energy of a fluid at any particular point above a datum plane is the sum of the elevation head, the pressure head, and the velocity head. Stated mathematically: H = Z e P ρ + V2 2g Eq 17-1 If there are no friction losses and no energy is added to or taken from the system, H is constant for any point in the fluid. In reality, whenever fluid is moving there is friction loss (h L ). This loss describes the difference in total energy at two points in the system. Expressing the energy levels at Point 1 versus Point 2 then becomes: Z el P 1 + V 2 1 ρ 1 2g = Z e P 2 + V 2 2 ρ 2 2g + h L Eq 17-2 All practical formulas for fluid flow are derived from the above. Modifications to Eq 17-2 have been proposed by many investigators to account for the friction losses. Fluid Physical Properties The physical properties of a flowing fluid must be known to predict pressure drop in piping. The two properties entering into the solution of most fluid flow problems are viscosity and density. Viscosity expresses the readiness with which a fluid flows when it is acted upon by an external force. Two types of viscosity measurements are used, absolute and kinematic. Absolute viscosity is a measure of a fluid s internal resistance to deformation or shear. Kinematic viscosity is the ratio of absolute viscosity to mass density. The absolute viscosity will be used for all calculations in this section. Viscosity is temperature dependent. The viscosity of most liquids decreases with an increase in temperature, whereas that of gases increases. Pressure has almost no effect on the viscosity of liquids or near perfect gases. On the other hand, the viscosity of saturated or slightly superheated vapors is changed appreciably by pressure changes. The viscosity of steam is readily available, but the viscosity of other vapors may not be known. µ L = liquid viscosity, cp µ n = mixture viscosity for Dukler calculation, cp ρ = single phase fluid density, lb/ft 3 ρ avg = average density, lb/ft 3 [ρ avg = 1/2 (ρ in + ρ out )] ρ a = air density at 60 F and 14.7 psia, lb/ft 3 ρ w = water density at 60 F and 14.7 psia, 62.4 lb/ft 3 ρ g = gas density, lb/ft 3 ρ L = liquid density, lb/ft 3 ρ k = two phase mixture density for Dukler calculation, lb/ft 3 σ = interfacial tension at flowing conditions, dyne/cm σ wa = interfacial tension of air and water at 60 F and 14.7 psia, 72.4 dyne/cm Specific volume is the inverse of density. Specific gravity of a liquid is the ratio of the density of the liquid at a specified temperature to the density of water at 60 F. The specific gravity of gas is defined as the ratio of the molecular weight of the gas to the molecular mass of air. MW (gas) S = Eq 17-3 MW (air) Flow in Pipes and Reynolds Number At low velocities, fluid molecules or particles carried by the fluid move in a reasonably straight line. Velocity of the fluid is maximum at the center of the pipe and zero at the pipe wall. This flow pattern is referred to as laminar. If the velocity is increased it will reach a critical point where fluid particles begin to show a random motion transverse to the direction of flow. This is the critical velocity. This random motion is typical of what is referred to as turbulent flow. Above the critical velocity the flow is considered to be completely turbulent even though there is always a boundary layer at the pipe wall where flow is laminar. In the turbulent zone the velocity profile is more nearly straight across the face of the pipe. Reynolds developed a dimensionless number that may be considered as the ratio of the dynamic forces of mass flow to the shear stress due to viscosity. The Reynolds number is: Re = DVρ Eq 17-4 µ e If the Reynolds number is less than 2000, flow may be considered laminar. If it is above 4000, the flow is turbulent. In the zone between 2000 and 4000 the flow could be either turbulent or laminar, but cannot be predicted by the Reynolds number. If a non-circular conduit is encountered, the Reynolds number can be approximated by using an equivalent diameter for D. The equivalent diameter would equal four (4) times the hydraulic radius. The hydraulic radius is defined as: Area of Flowing Fluid Hydraulic Radius = Eq 17-5 Wetted Perimeter This conversion would not apply to extremely narrow shapes where the width is small relative to the length. In such cases an approximation may be used wherein one-half the width of the passage is equal to the hydraulic radius. 17-2

3 Pressure Loss Due to Friction Flow is always accompanied by friction. This friction results in a loss of energy available for work. A general equation for pressure drop due to friction is the Darcy-Weisbach 2 (often referred to as simply the Darcy) equation. This equation can be rationally derived by dimensional analysis, with the exception of the friction factor, f m, which must be determined experimentally. Expressed in feet of fluid this equation is: h L = f m L V 2 2 g D Eq 17-6 Converting to pounds per square inch, the equation becomes: P f = ρ f m L V 2 (144) D (2g c ) Eq 17-7 It should be noted that the Moody friction factor 3, f m, is used in the equations above. Some equations are shown in terms of the Fanning friction factor, f f, which is one fourth of f m (f m = 4.0 f f ). A graph of both Fanning and Moody friction factors as a function of Reynolds number appears in Fig The Darcy-Weisbach equation is valid for both laminar and turbulent flow of any liquid, and may also be used for gases with certain restrictions. When using this equation, changes in elevation, velocity, or density must be accounted for by applying Bernoulli s theorem. The Darcy-Weisbach equation must be applied to line segments sufficiently short such that fluid density is essentially constant over that segment. The overall pressure drop is the sum of the P f values calculated for the individual segments. For gas applications the segmental length may be relatively short, as compared to liquid applications, since many gas applications involve compressible gases where gas densities vary with pressure. Friction Factor and Effect of Pipe Roughness When the fluid flow is laminar (Re<2000), the friction factor has a direct relationship to the Reynolds number, such that: f m = 64 /Re or f f = 16 /Re Eq 17-8 Pipe roughness has no effect on the friction factor in laminar flow. Substitution of the formula for Reynolds number, Eq 17-4, into Eq 17-8, yields the following: f m = 64 µ e DVρ = 64 µ 12 Vρ 1488 d Eq 17-9 This expression can then be substituted for the friction factor in Eq 17-7, resulting in the following formula for pressure loss in pounds per square inch: P f = µlv d 2 Eq Eq is commonly known as Poiseuille s law for laminar flow. When the flow is turbulent, the friction factor depends on the Reynolds number and the relative roughness of the pipe, FIG Friction Factors

4 ε/d, which is the roughness of the pipe, ε, over the pipe diameter, D. Fig incorporates the relative roughness of the pipe into the determination of the friction factor. Fig indicates relative roughness and friction factors for various piping materials. These figures are based on the iterative solution of the following equation developed by Colebrook. 4 1 = 2 log ε 10 f m 3.7 D Eq Re f m Equivalent Length of Valves and Fittings The pressure drop effects of valves and fittings can be accounted for by addition of the "equivalent lengths" of the fittings to the actual piping lengths. This augmented pipe length is then used in any of the following pressure drop calculation techniques. A table of equivalent lengths for a number of representative valves and fittings appears in Fig Compressibility of Gases For more accurate values of Z, refer to Section 23. For more approximate calculations, the value of the average compressibility factor, Z avg, may be calculated from the following equations: and Z avg = 1 (F pv ) 2 Eq F pv = 1 + (P avg ) (3.444 ) (10 5 ) (10 (1.785) (S) ) Eq T avg Fig contains a plot of the deviation factor, F pv, virtually identical to those calculated by this equation. An estimate for Z avg at pressures below 100 psi is: Z avg = P avg Eq SINGLE PHASE FLOW Transmission Line Gas Flow Isothermal Flow The steady-state, isothermal flow behavior of gas in pipelines is defined by a general energy equation of the form: Q = T b P E 1 P P 2 b f f S L m T avg Z avg 0.5 d 2.5 Eq This equation is completely general for steady-state flow, and adequately accounts for variations in compressibility factor, kinetic energy, pressure, and temperature for any typical line section. However, the equation as derived involves an unspecified value of the transmission factor, 1/f f. The correct representation of this friction factor is necessary to the validity of the equation. The friction factor is fundamentally related to the energy lost due to friction. In the derivation of the general energy equation, all irreversibilities and non-idealities, except for those covered by the real gas law, have been collected into the friction loss term. Empirical methods historically and currently used to calculate or predict the flow of gas in a pipeline are the result of various correlations of the transmission factor substituted into the general energy equation. Examination of the relationships presented by various authors shows that their forms differ primarily in the inherent or specified representation of the transmission factor which defines the energy lost in resistance to flow for various pipe sizes, roughnesses, flow conditions, and gases. To obtain Eq 17-15, which is convenient for general calculations, a number of simplifying assumptions have been made. For other than pipeline sections with a very high pressure gradient, the change in the kinetic energy of the gas is not significant, and is assumed equal to zero. It is also assumed that the gas temperature is constant at an average value for the section considered; the compressibility factor is constant at the value characterized by the average gas temperature and pressure; and in the term giving the effect of elevation change, the pressure is constant at the average value. In the range of conditions to which pipeline flow equations are ordinarily applied, averages are usually sufficiently accurate. Average temperatures are calculated as indicated in Fig The average pressure in the line can be computed by: P avg = 2 3 P 1 + P 2 P 1 P 2 P 1 + P Eq In the absence of field data indicating otherwise, an efficiency factor, E, of 1.0 is usually assumed. The AGA Equations The AGA Equations were developed to approximate partially and fully turbulent flow using two different transmission factors. The fully turbulent flow equation accounts for the relative pipe roughness, ε/d, based on the rough-pipe law. 4 This equation uses the following transmission factor: 1/f f = 4 log 3.7 D 10 ε Eq When the transmission factor for fully turbulent flow is substituted in the general energy equation (Eq 17-15), the AGA Equation for fully turbulent flow becomes: Q = T b P E 4 log 3.7 D P P 2 10 b ε S L m T avg Z avg 0.5 d 2.5 Eq The partially turbulent flow equation is based on the smooth-pipe law 4 and is modified to account for drag-inducing elements. The transmission factor for this equation is: 1/f f = 4 log R e Eq /f f Substituting 1/f f from Eq into Eq does not provide an equation which can be solved directly. For partially turbulent flow a frictional drag factor must also be applied to account for the effects of pipe bends and irregularities. These calculations are beyond the scope of this book and the AGA "Steady Flow in Gas Pipelines" 6 should be consulted for a detailed treatment of partially turbulent flow. The Weymouth Equation The Weymouth Equation, published in , evaluated the coefficient of friction as a function of the diameter. f f = d 1/3 Eq /f f = d 1/6 Eq When the friction factor, f f, is substituted in the general energy equation, Weymouth s Equation becomes: 17-4

5 FIG Relative Roughness of Pipe Materials and Friction Factors for Complete Turbulence

6 FIG Equivalent Length of Valves and Fittings in Feet Nominal Pipe size in. Globe valve or ball check valve Angle valve Swing check valve Plug cock Gate or ball valve Welded 45 ell Threaded Short rad. ell Welded Long rad. ell Hard T Soft T 90 miter bends Enlargement Contraction Sudden Std. red. Sudden Std. red. Equiv. L in terms of small d Threaded Welded Threaded Welded Threaded Welded Threaded 2 miter 3 miter 4 miter d/d = 1/4 d/d = 1/2 d/d = 3/4 d/d = 1/2 d/d = 3/4 d/d = 1/4 d/d = 1/2 d/d = 3/4 d/d = 1/2 d/d = 3/4 Q = (433.5) T b P E P P 2 b S L m T avg Z avg 0.5 d Eq The Weymouth formula for short pipelines and gathering systems agrees more closely with metered rates than those calculated by most other formulae. However, the degree of error increases with pressure. If the Q calculated from the Weymouth formula is multiplied by 1/Z, where Z is the compressibility factor of the gas, the corrected Q will closely approximate the metered flow. Fig shows a plot of the deviation factor, 1/Z, of a common gas and can be used safely if exact data is not available. The equation cannot be generally applied to any variety of diameters and roughness, and in the flow region of partially developed turbulence, it is not valid. The Weymouth Equation may be used to approximate fully turbulent flow by applying correction factors determined from the system to which it is to be applied. Graphs showing gas flow calculations based on the Weymouth equation are shown in Fig. 17-6a and 17-6b. Panhandle A Equation In the early 1940s Panhandle Eastern Pipe Line Company developed a formula for calculation of gas flow in transmission lines which has become known as the Panhandle A Equation. This equation uses the following expressions of Reynolds number and transmission factor. Re = QS d Eq /f f = QS d = (Re) Eq The transmission factor assumes a Reynolds number value from 5 to 11 million based on actual metered experience. Substituting Eq for 1/f f in the general energy equation (Eq 17-15), the Panhandle A Equation becomes: Q = T b P E P P 2 d b S L m T avg Z avg Eq This equation was intended to reflect the flow of gas through smooth pipes. When "adjusted" with an efficiency factor, E, of about 0.90, the equation is a reasonable approximation of the partially turbulent flow equation. The equation becomes less accurate as flow rate increases. Many users of the Panhandle A Equation assume an efficiency factor of Panhandle B Equation A new or revised Panhandle Equation was published in This revised equation is known as the Panhandle B Equation and is only slightly Reynolds number dependent. Therefore, it more nearly approximates fully turbulent flow behavior. The transmission factor used here is: /f f = QS d = (Re) Eq Substituting Eq for 1/f f in the general energy equation (Eq 17-15), the Panhandle B Equation becomes: Q = 737 T b P 1.02E P P 2 d b 2.53 Eq S L m T avg Z avg The equation can be adjusted through the use of an efficiency term that makes it applicable across a relatively limited range of Reynolds numbers. Other than this, however, there are no means for adjustment of the equation to correct it for variations in pipe surface. Adjusted to an average flowing Reynolds number, the equation will predict low flow rates at 17-6

7 FIG Deviation Factors 8 Note: Refer to Section 23 for more accurate compressibility factors. low Reynolds numbers, and high flow rates at high Reynolds numbers, as compared to a fully turbulent flow equation. Efficiencies based on the Panhandle B equation decrease with increasing flow rate for fully turbulent flow. The efficiency factor, E, used in the Panhandle B equation generally varies between about 0.88 and Conclusions The successful application of these transmission line flow equations in the past has largely involved compensation for discrepancies through the use of adjustment factors, usually termed "efficiencies." These efficiencies are frequently found in practice by determining the constant required to cause predicted gas equation behavior to agree with flow data. As a result, the values of these factors are specific to particular gas flow equations and field conditions and, under many circumstances, vary with flow rate in a fashion that obscures the real nature of flow behavior in the pipe. The Reynolds number dependent equations, such as the Panhandle equations, utilize a friction factor expression which yields an approximation to partially turbulent flow behavior in the case of the Panhandle A equation, and an approximation to fully turbulent behavior in the case of the Panhandle B. These equations suffer from the substitution of a fixed gas viscosity value into the Reynolds number expression, which, in turn, substituted into the flow equation, results in an expression with a preconditioned bias. Regardless of the merits of various gas flow equations, past practices may dictate the use of a particular equation to maintain continuity of comparative capacities through application of consistent operating policy. A summary of comparisons between transmission factors used in the above gas equations are shown in Fig Reference should be made to "Steady Flow in Gas Pipelines" 6, published by American Gas Association, for a complete analysis of steady flow in gas pipelines. Low Pressure Gas Flow Gas gathering often involves operating pressures below 100 psi. Some systems flow under vacuum conditions. For these low pressure conditions, equations have been developed that give a better fit than the Weymouth or Panhandle equations. Two such formulas are: The Oliphant Formula 9 for gas flow between vacuum and 100 psi: Q = 42 (24) d d T b 30 P b P P 2 S T Eq L m The Spitzglass Formula for gas flow below 1 psig at 60 F: 1 2 Q = (24) (3550) (P 1 P 2) d 5 SL d d Plant Piping Gas Flow 1 2 Eq For estimating pressure drop in short runs of gas piping, such as within plant or battery limits, a simplified formula for compressible fluids is accurate for fully turbulent flow, assuming the pressure drop through the line is not a significant fraction of the total pressure (i.e. no more than 10%). The following method is a simplification of the Darcy formula, which eliminates calculation of f m, the Moody friction factor. This simplification was checked over a wide range of flows and densities for pressure drops of 0.25 to 1.5 psi/100 ft. Density was varied over a range of 100 to 1; flows varied over a range of 75 to 1. Pressure variation was from atmospheric to 1000 psia. The error from using the simplified approach as compared to the actual friction factor calculated in the Darcy formula was from zero to 5%, with the simplified approach giving consistently lower calculated pressure drop for a given flow. The Darcy formula can be written in the simple form: P 100 = W f m ρ d 5 Eq Simplifying, C 1 = W 2 (10 9), and C 2 = 336,000 f m, then d 5 P 100 = C 1 C 2 ρ Eq C 1 = ( P 100) ρ C 2 = discharge factor from chart, Fig C 2 = ( P 100) ρ C 1 = size factor Fig

8 FIG. 17-6a Gas Flow Based On Weymouth Formula 17-8

9 FIG. 17-6b Gas Flow Based On Weymouth Formula 17-9

10 FIG Comparison of Gas Equation Transmission Factors for Nominal 20 Inch Pipe C 2 incorporates the friction factor, assuming clean steel. Using this simplified approach, new lines can be sized by setting the desired P 100 and solving for C 2 with a given flow. For a given flow and pipe size, P 100 can be solved directly. Example 17-1 Calculate the pressure drop in a 10-in., Schedule 40 pipe for a flow of 150,000 lb/hr of methane. Temperature is 60 F and pressure is 750 psia. The compressibility factor is (from Fig. 23-3). Solution Steps ρ = (750) ( ) (0.905) C 1 from Fig is 22.5 C 2 from Fig is P 100 = C 1C 2 ρ = 22.5 (0.0447) 2.38 = 2.38 lb /ft3 = psi/100 ft using Eq Example 17-2 Calculate the required line size (of Schedule 40 pipe) to give P 100 = 1 psi or less when flowing 75,000 lb/hr of methane at 400 psia and 100 F. The compressibility factor is 0.96 (from Fig. 23-8). Solution Steps ρ = (400) ( ) (0.96) = 1.11 lb/ft3 C 1 from Fig is 5.6 C 2 = ( P 100) ρ C 1 = 1 (1.11) 5.6 = 0.20 From Fig the smallest size of Schedule 40 pipe with C 2 less than 0.20 is 8-in. pipe. For 8 in. Sch 40 pipe, C 2 is The actual pressure drop can then be calculated as: P 100 = 5.6 (0.146) 1.11 = 0.74 psi/100 ft using Eq for the above flow conditions. Liquid Flow For the calculation of pressure drop in liquid lines, the Darcy-Weisbach method, Eq 17-6, can be used. The calculation is simplified for liquid flows since the density can reasonably be assumed to be a constant. As a result, the Darcy-Weisbach calculation can be applied to a long run of pipe, rather than segmentally as dictated by the variable density in gas flow. In addition, several graphical aids are available for pressure drop calculation. Elevation pressure drops must be calculated separately using Eq These elevation pressure gains or losses are added algebraically to the frictional pressure drops. P e = ( ) ρ L Z e Eq Water A graph showing pressure drop for water per 100 feet as a function of flow rate in gallons per minute and pipe size is shown in Fig These data are based on the Hazen and Williams empirical formula 10 using a "C" constant of

11 FIG Simplified Flow Formula for Compressible Fluids 5 Values of C 1 Hydrocarbon A graph showing pressure drop for hydrocarbons per 100 feet as a function of flow rate in gallons per minute and pipe size is shown in Fig This graph assumes a specific gravity of 1.0 (water). To correct for different liquid densities, the value read from Fig must be multiplied by the actual specific gravity to obtain the correct pressure loss. Steam Flow Fig contains a graphical representation of Fritzsche s formula 11 for calculating pressure drop in steam lines. Fritzsche s formula and instructions for the chart usage are given in Fig The Babcock formula 5 for steam flow is: P f = 3.63 (10 8 ) d W 2 L d 6 ρ Fire Stream Flow Eq Fig is a table permitting rapid computation of the behavior of various sized fire nozzles. The table also includes the estimated pressure drop in 100 feet of inch diameter fire hose. TWO PHASE FLOW Two-phase flow presents several design and operational difficulties not present in single phase liquid or vapor flow. Frictional pressure drops are much harder to calculate. For cross-country pipelines, a terrain profile is needed to calculate elevation pressure drops. At the downstream end of a pipeline, it is frequently necessary to separate the liquid and vapor phases in a separator. The presence of liquid slugs complicates this process, and a slug catcher may be required. Flow Regime Determination Several empirical flow regime maps have been presented that determine vapor-liquid flow patterns as a function of fluid properties and flow rates. Diagrams of these flow patterns are shown in Fig One map commonly used was developed by Gregory, Aziz, and Mandhane 12 for horizontal flow. This map appears as Fig The coordinates of the map are: V sl = superficial liquid velocity = Q L /A Eq V sg = superficial gas velocity which is commonly used for design purposes in welded and seamless steel pipe. Hazen and Williams formula for flow of water: q = d 2.63 C P 1 P 2 L Where: C = 140 for new steel pipe 0.54 C = 130 for new cast iron pipe Eq C = 100 is often used for design purposes to account for pipe fouling, etc. = Q g /A Eq Mandhane proposed a fluid property correction to the superficial velocities, but concluded that the fluid property effects are insignificant compared to the errors in the empirical map. The map reports the flow regimes: stratified, wavy, annular mist, bubble, slug, and dispersed. Care should be taken in the interpretation of these flow maps as the regime boundaries are strongly affected by pipe inclination. In particular, horizontal flow regime maps must not be used for vertical flow, and vertical flow regime maps must not be used for horizontal flow. The Mandhane map given in Fig was developed for horizontal lines flowing air and water at near atmospheric pressure. Inclinations in the range of degrees can cause substantial regime boundary movement. In addition, flow regime boundary adjustment has been observed due to fluid pressure, pipe diameter, and surface tension. 13,22 The gas density increase caused by high pressure acts to move the slugmist boundary to lower superficial gas velocities, while increased pipe diameter acts to increase the stratified wavy flow regime at the expense of the slug flow regime. In addition, foamy fluids having a high surface tension have been observed to flow in the dispersed flow regime even though Mandhane 17-11

12 FIG Simplified Flow Formula for Compressible Fluids 5 Values of C 2 Nominal pipe size in. Schedule number Value of C s x s x s x s x xx s x xx s x xx s x xx s x xx s x xx s x xx s x xx s x s x xx s x xx 4.93 Nominal pipe size in. Schedule number 6 40 s x xx s x Value of C xx s x s x s x s Nominal pipe size in. Schedule number Value of C 2 40 x s x s x s x Note: The letters s, x, and xx in the columns of Schedule Numbers indicate Standard, Extra Strong, and Double Extra Strong pipe respectively

13 FIG Pressure Drop for Flowing Water 17-13

14 FIG Pressure Drop for Hydrocarbon Liquids in Smooth Pipe would have predicted superficial liquid velocities too low to cause dispersed flow. A flow regime map generated by Taitel and Dukler 13 contains explicit inclination effects and should be used for inclined pipes. This latter method also contains explicit pipe diameter effects but lacks any way of accounting for surface tension. For vertical flow, the stratified flow regime cannot exist as there is no preferred direction for the liquid to settle. An empirical flow regime map developed by Aziz 14 for vertical upward flow is shown in Fig The coordinates for this flow map are the same as for the Mandhane map in Fig except that fluid property corrections are used. The coordinates used in the Aziz vertical map are: N x = V sg X A Eq N y = V sl Y A Eq X A = ρ g ρ Y A Eq a Y A = ρ L σ wa ρ w σ 0.25 Eq For pipe inclinations greater than degrees, flow regime patterns resemble those of vertical flow more than those of horizontal flow, and the Aziz vertical map should be used. For vertical downward flow, the flow patterns can be more complicated. A generally accepted vertical down-flow map is not available. Although the designer frequently does not have the choice, avoidance of the slug flow regime in horizontal flow and the slug and froth flow regimes in vertical flow is desirable. Slug flow introduces a flow rate and pressure intermittency that may be troublesome to process control. In some cases, slug flow may be avoided by the choice of smaller pipe sizes. Of course, frictional pressure drop may be increased by use of this smaller pipe. In vertical upflow, oversizing the pipe may result in a bubble flow regime, and a large liquid inventory. This liquid inventory may cause excessive hydrostatic presure drops. Example 17-3 A vapor-liquid mixture is flowing vertically upward in a pipe having an inside diameter of 8.0 inches. The fluid is a hydrocarbon liquid-hydrocarbon vapor mixture. The liquid density is 52 lb/ft 3 and the vapor density is 2.0 lb/ ft 3. The interfacial surface tension is 20 dynes/cm. The liquid volumetric flow rate is 0.17 ft 3 /sec and the vapor flow rate is 17-14

15 FIG Pressure Drop in Steam Pipes by Fritzsche s Formula

16 FIG Table of Effective Fire Streams Smooth nozzles, size 3 4 " 7 8 " 1" Pressure at hydrants, psi Pressure at nozzle, psi Pres. lost in 100 ft in. hose psi Vertical height, ft Horizontal distance, ft Gal discharge/ min Smooth nozzles, size " " " Pressure at hydrants, psi Pressure at nozzle, psi Pres. lost in 100 ft in hose psi Vertical height, ft Horizontal distance, ft Gal discharge/min Using 100 ft of in. ordinary best quality rubber lined hose between nozzle and hydrant or pump. The vertical and horizontal distance in above table are "effective" streams. The "effective stream" is one that has not broken up into a spray and which will project three fourths of the water through a circle 10 in. in diameter, at the distance indicated. The bulk of the stream and extreme drops will carry a greater distance, but the stream is spread out too much to penetrate a hot fire and reach the burning materials before evaporation. The vertical height "h" in feet, which the bulk of the stream will carry, may be computed by formula h = 0.91 V2 2g The gallons/minute discharge in the above table check approximately with the formula: V = C 2 gh = gpm A V A then gpm = V = Velocity in feet per second g = Acceleration due to gravity, = ft/sec A = Area of nozzle, in square inches C = Coefficient for smooth nozzles = 0.98 H = Head in feet = pressure x 2.31 Reference: Fire Protection Handbook, 14th Edition, 1976, National Fire Protection Association FIG Two Phase Flow Regimes FIG Mandhane et al. Horizontal Flow Regime Map 12 Bubble Slug Annular Mist Stratified Wave 17-16

17 FIG Aziz et al. Vertical Up-Flow Regime Map 14 can serve as a basis for hand calculation generated by Dukler 19 and an elevation pressure drop correlation by Flanigan. 20 Frictional Component 18 Using the Dukler frictional pressure drop calculation method, the frictional pressure drop is given by the equation: P f = f n f tpr ρ k V m 2 L m ( )d where and Eq ρ k = ρ L λ 2 H Ld + ρ g (1 λ) 2 (1 H Ld) Eq ft 3 /sec measured at acutal conditions. What flow regime is to be expected? Solution Steps Y A = (52.0) (72.4) (62.4 (20) = 1.32 using Eq X A = (2.0) (1.32) = 3.91 using Eq V sg = = 1.43 ft/sec using Eq (π/4) (8/12) V sl = 0.17 = 0.49 ft/sec using Eq (π/4) (8/12) N x = (1.43) (3.91) = 5.60 ft/sec using Eq N y = (0.49) (1.32) = 0.64 ft/sec using Eq Fig shows that this flow is in the slug flow regime. Pressure Drop Calculation Calculation of pressure drop in two-phase flow lends itself better to computer calculation than to hand calculation. Several two-phase pressure drop correlations are available for both horizontal and vertical flows. 15,16,17 Due to the complexity of two-phase flow, uncertainties associated with pressure drop calculations are much greater than uncertainties in singlephase pressure drop calculations. As a result, errors in calculated two-phase pressure drops in the order of plus or minus twenty percent may normally be anticipated, especially in circumstances where fluid velocities are unusually high or low, where terrain is rugged, or where fluid properties are inadequately known. In addition, different two-phase flow correlations may give significantly different pressure drops. In order to evaluate these differences, several correlations should be used. A method suggested by the American Gas Association 18 λ = Q L Q L + Q g Eq The single phase friction factor, f n, can be obtained from the correlation: 19 f n = (Re y ) 0.32 Eq The mixture Reynolds number, Re y, is calculated according to the equation: Re y = (124.0) ρ k V m d µ n Eq Calculation of this Reynolds number requires determination of mixture velocity, V m, and mixture viscosity, µ n. These quantities can be determined according to: V m = V sl + V sg Eq µ n = µ L λ + µ g (1 λ) Eq The two-phase friction factor ratio, f tpr, representing a twophase frictional "efficiency" can be determined by reference to Fig or by the equation: f tpr = 1 + y y y y y 4 Eq where y = ln(λ). The remaining quantity to be calculated in the Dukler scheme is an estimate of the liquid holdup, H Ld. This holdup can be estimated using Fig This figure gives liquid holdup as a function of λ and Re y. Since Re y is itself a function of liquid holdup, the calculation is, in general, iterative. For most calculations, however, the Re y line can be used for a first estimate. Elevation Component The elevation component of pressure drop can be found using the Flanigan method. In this method, the elevation component is calculated using the equation: P e = ρ L H Lf 144 Σ Z e Eq where H Lf is determined from Fig or calculated according to the formula: H Lf = (V sg ) Eq The term Z e is the vertical elevation rise of a hill. The rises are summed. No elevation drops are considered. This is tantamount to ignoring any possible hydrostatic pressure recoveries in downhill sections of pipeline and may lead to a considerable error in the pressure drop analysis

18 Once the frictional component or pressure drop is found using the Dukler method, and the elevation component is found using the Flanigan method, the overall two-phase pressure drop is found by summing the friction and elevation components. P t = P e + P f Eq Since fluid properties and liquid holdups can change rapidly in a two-phase line, accuracy is improved if this AGA calculation procedure is performed segmentally. The need for segmental calculations is one of the reasons why two-phase calculations are best suited for computer calculation. Liquid Holdup The liquid holdup correlation given in Fig is intended only for use in the Dukler friction pressure drop calculation. A correlation by Eaton et al. 21 is better suited for liquid holdup determination in liquid inventory calculations. The Eaton 21 holdup correlation is shown in Fig In this figure, the holdup fraction, H Le, is plotted directly as a function of the dimensionless group, N E. This dimensionless group is of the form: 1.84 (N Lv ) P avg (N L ) 0.1 P b N e = Eq N gv (N d ) where N Lv = V ρ L sl σ N gv = 1.938V ρ L sg σ N d = d ρ L σ N L = µ L ρ L σ 3 Eq Eq Eq Eq The Eaton correlation has been found reasonably accurate by several investigators, particularly for low holdup flows. The liquid holdup fraction, H Le, is the fraction of the flow area of the pipe occupied by liquid. To calculate the liquid inventory in the pipe, I L, the pipe internal volume is multiplied by this holdup fraction. I L = (28.80) H Le d 2 L m Eq Since holdup fractions may change along the length of the pipe, a segmental calculation is more accurate. Example 17-4 A pipeline segment with a 6-inch inside diameter, 0.75 miles long, transports a mixture of gas and oil. The pipeline has a gradual upward slope and rises 100 feet over the 0.75 mile length. The inlet pressure of the pipeline is 400 psia, liquid viscosity is 20 cp, the vapor viscosity is cp, and the interfacial surface tension is 15 dynes/cm. The liquid flow rate is 10 ft 3 /min and the vapor flow rate is 250 actual ft 3 /min. The density of the liquid phase is 55 lb/ft 3, and the density of the gas phase is 1.3 lb/ft 3 at operating conditions. What is the pressure at the downstream end of the line segment, and what is the liquid inventory of the line? Solution Steps Calculate the flowing liquid volume fraction using Eq λ = = Calculate the mixture viscosity, µ n using Eq µ n = (20) (0.038) + (0.015) ( ) = cp For a first guess, assume H Ld = λ and estimate ρ k using Eq ρ k = (55) (0.038) (1.3) ( )2 ( ) = lb/ft 3 Calculate the superficial velocities and the mixture velocity V sl = = ft/sec using Eq (π/4) (6/12) 2 (60) V sg = 250 = ft/sec using Eq (π/4) (6/12) 2 (60) V m = = ft/sec using Eq Calculate an estimate of the mixture Reynolds number, Re y, using Eq Re y = (124.0) (3.341) (22.07) (6.0) (0.774) = 70,878 From Fig , determine a better estimate for the holdup fraction H Ld using λ = 0.038, Re y = 70,878 H Ld = 0.12 Using this improved H Ld, recalculate ρ k using Eq ρ k = (55) (0.038)2 (0.12) + (1.3) ( ) = lb/ft 3 Using this improved ρ k recalculate Re y using Eq Re y = (124.0 ) (2.029 ) (22.07) (6.0) (0.774) = 43,044 From Fig with λ = and Re y = 43,044, H Ld = Another iteration using H Ld = 0.16 indicates Re y = 40,923 and H Ld = Calculate the single phase friction factor with Re y = 40,923, using Eq f n = (40,923) = Determine the two-phase friction factor, f tpr, from Fig using λ= f tpr = 2.59 Now for λ = 0.038, H Ld = 0.16 then ρ k = using Eq Calculate the frictional component of pressure drop P f using Eq P f = (0.0223) (2.59) (1.929) (22.07)2 (0.75) ( ) (6.0) Find H Lf from Fig using V sg = ft/sec H Lf = 0.13 = psi Determine the elevation component of pressure drop, P e, using Eq P e = (55) (0.13) (100) (144) = 4.97 psi Find the total pressure drop, P t, using Eq P t = = psi Find the segment discharge pressure P 2 = = psia 17-18

19 FIG Two-Phase Friction Factor Ratio 19 FIG Liquid Holdup Correlation

20 FIG Flanigan Liquid Holdup Correlation 20 The pipeline segment has a discharge pressure of psia. To calculate the liquid inventory, the liquid holdup fraction from Eaton s correlation must be found. First determine the nondimensional parameters: FIG Eaton Liquid Holdup Correlation 21 N Lv = (1.938) (0.849) (55/15) 0.25 = using Eq N gv = (1.938) (21.22) (55/15) 0.25 = using Eq N d = (10.073) (6.0) (55/15) 0.5 = using Eq N L = ( ) (20.0) (55) (15) 3 = using Eq Determine Eaton s nondimensional abscissa, N E, using Eq N E = (1.84) (2.277)0.575 (400/14.73) 0.05 (0.152) 0.1 (56.91) (115.73) = From Fig , read the holdup fraction, H Le H Le = 0.14 Note that this estimate is close to the H Lf predicted in Fig for elevation pressure drop determination. It also coincides closely with the value of 0.16 from Fig Calculate the pipeline segment liquid inventory from Eq I L = (28.80) (0.14) (6.0) 2 (0.75) = ft 3 The pipeline segment contains cubic feet of liquid at any instant. Liquid Slugging Purpose of Separators The slug flow regime is frequently encountered for pipe sizes and flow rates used in process and transmission piping. Liquid slugging introduces an additional design and operational difficulty as liquid and vapor must generally be separated at the downstream end of the two-phase flow line. The downstream separator serves both as a liquid-vapor disengaging device and as a surge vessel to absorb the fluctuating liquid flow rates caused by slugging. In order to size the separator or slug catcher, the length of the incoming slugs must be determined. Slug length calculation 17-20

21 INLET LIQUID OUTLET FIG Multiple Pipe Slug Catcher FIG Example Line Drip GRADE GAS FLOW SLOPED DRIP BELOW LINE VAPOR OUTLET DRIP VESSEL DRIP LIQUID DRIP VALVE NO methods are not well developed, and there is large uncertainty in slug length determination. Mechanisms of Slug Generation Liquid slug lengths are difficult to determine in part because there are at least four identifiable mechanisms for liquid slug generation. Slugs can form as the result of wave formation at the liquid-gas interface in a stratified flow. When the liquid waves grow large enough to bridge the entire pipe diameter, the stratified flow pattern breaks down and a slug flow is formed. Slugs can also form due to terrain effects. Liquid collects at a sag in the pipeline and blocks the gas flow. The pressure in this blocked gas rises until it blows the accumulated liquid in the sag out as a slug. Changes in pipeline inlet flow rate can also cause slugs. When the inlet flow rate increases, the liquid inventory in the pipeline decreases, and the excess liquid forms a slug or series of slugs. Finally, pigging can cause very large liquid slugs as the entire liquid inventory of the line is swept ahead of the pig. Of the four mechanisms described, wave growth normally produces the shortest slugs, followed in length by terrain generated slugs. Methods for calculating wave induced slugs were described by Greskovich and Shrier 22, and by Brill et al. 23 A preliminary scheme for calculating terrain generated slugs was reported by Schmidt. 24 Analytical methods for determining inlet flow rate generated slugs were given by Cunliffe, 25 and a method of analyzing pigging dynamics was given by McDonald and Baker. 26 Slug Catchers Slug catchers are devices at the downstream end or other intermediate points of a pipeline to absorb the fluctuating liquid inlet flow rates through liquid level fluctuation. Slug catchers may be either a vessel or constructed of pipe. All size specifications discussed in Section 7 to provide residence time for vapor-liquid disengagement also apply to vessels used as slug catchers. In addition, sufficient volume must be provided for liquid level fluctuation. Particularly for high pressure service, vessel separators may require very thick walls. In order to avoid thick wall vessels, slug catchers are frequently made of pipe. Lengths of line pipe tens or hundreds of feet long are used as long, slender horizontal separators. The pipe is generally inclined from one to ten degrees and banks of these slightly inclined pipes are frequently manifolded together. Pipe type slug catchers are frequently less expensive than vessel type slug catchers of the same capacity due to thinner wall requirements of smaller diameter pipe. The manifold nature of multiple pipe slug catchers also makes possible the later addition of additional capacity by laying more parallel pipes. A schematic of a multiple pipe (harp) slug catcher appears in Fig Different pipe inclinations and different manifolding arrangements are favored by different designers. An example of a line drip catcher is shown in Fig A drip vessel is connected to the incoming pipeline and often laid beneath it. A flow line from the drip vessel is used to blow the liquids out to a storage or surge vessel as they accumulate. Pigging Pipelines are pigged for several reasons. If water is present in the line, it must be removed periodically in order to minimize corrosion. This water accumulates in sags in the pipeline, and these low spots are particularly susceptible to corrosion. Pipelines are also pigged to improve pressure drop-flow rate performance. Water or hydrocarbon liquids that settle in sags in the pipeline constitute partial blockages that increase pressure drop. Pigging can remove these liquids and improve pipeline efficiency. Pigging can also be used as a means of limiting the required slug catcher size. By pigging at frequent intervals, liquid inventory buildup in a pipeline can be reduced, and the maximum slug size can be limited. The required downstream slug catcher size must take into account pigging frequency. Operational hazards are associated with pigging. The very large slugs swept ahead of the pig may overwhelm inade